Proving Non-Existence of Limit of Function f: (0,1) --> R | Negation Method

[renolovexoxo]

Homework Statement

Negate the definition of the limit of a function, and use it to prove that for the function
f : (0; 1) --> R where f(x) 1/x, lim x-->0 f(x) does not exist.

Homework Equations

The limit of f at a exists if there exists a real number L in R such that for every e>0 there exists d>0 such that for every x in the interval with 0<|x-a|<d then |f(x)-L|<e.

The Attempt at a Solution

The limit of f at a does not exist if for all real numbers L in R such that for every e>0 there exists d>0 such that there exists xin the interval with 00<|x-a|<d then |f(x)-L|>e

 

[SammyS]

Homework Statement

Negate the definition of the limit of a function, and use it to prove that for the function
f : (0; 1) --> R where f(x) 1/x, lim x-->0 f(x) does not exist.

Homework Equations

The limit of f at a exists if there exists a real number L in R such that for every e>0 there exists d>0 such that for every x in the interval with 0<|x-a|<d then |f(x)-L|<e.

The Attempt at a Solution

The following is true even if L is the limit at a .

The limit of f at a does not exist if for all real numbers L in R such that for every e>0 there exists d>0 such that there exists xin the interval with 0<|x-a|<d then |f(x)-L|>e

For every δ > 0 there needs to be an ε > 0 (this ε usually depends upon the δ) such that there is some x₀ for which 0 < |x₀-a| < δ and |f(x₀)-L|> ε .